#274714

order	512
nilpotency class	3
isoclinism	isoclinic class of #267051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,22,13,17,12,16)(11,18,19,15,21,23,20,24)(25,26,29,34,28,33,30,32)(27,35,38,40,37,31,36,39);(2,5)(6,8)(25,27)(26,31)(28,37)(29,38)(30,36)(32,40)(33,35)(34,39);(1,3,4,7)(2,6,5,8)(9,11,13,21)(10,15,17,24)(12,19,14,20)(16,23,22,18)(26,32)(27,36)(29,30)(31,35)(33,34)(37,38);(2,5)(6,8)(9,12,13,14)(10,16,17,22)(11,19,21,20)(15,23,24,18)(26,33)(31,35)(32,34)(39,40)

Polycyclic presentation:

2^9|a^2=fhi,c^2=eh,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181910);